0=6.28r^2+215073.2r-25722.88

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Solution for 0=6.28r^2+215073.2r-25722.88 equation:


Simplifying
0 = 6.28r2 + 215073.2r + -25722.88

Reorder the terms:
0 = -25722.88 + 215073.2r + 6.28r2

Solving
0 = -25722.88 + 215073.2r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 25722.88 = 25722.88
25722.88 + -215073.2r + -6.28r2 = -25722.88 + 215073.2r + 6.28r2 + 25722.88 + -215073.2r + -6.28r2

Reorder the terms:
25722.88 + -215073.2r + -6.28r2 = -25722.88 + 25722.88 + 215073.2r + -215073.2r + 6.28r2 + -6.28r2

Combine like terms: -25722.88 + 25722.88 = 0.00
25722.88 + -215073.2r + -6.28r2 = 0.00 + 215073.2r + -215073.2r + 6.28r2 + -6.28r2
25722.88 + -215073.2r + -6.28r2 = 215073.2r + -215073.2r + 6.28r2 + -6.28r2

Combine like terms: 215073.2r + -215073.2r = 0.0
25722.88 + -215073.2r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2
25722.88 + -215073.2r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
25722.88 + -215073.2r + -6.28r2 = 0.00

Begin completing the square.  Divide all terms by
-6.28 the coefficient of the squared term: 

Divide each side by '-6.28'.
-4096 + 34247.32484r + r2 = 0

Move the constant term to the right:

Add '4096' to each side of the equation.
-4096 + 34247.32484r + 4096 + r2 = 0 + 4096

Reorder the terms:
-4096 + 4096 + 34247.32484r + r2 = 0 + 4096

Combine like terms: -4096 + 4096 = 0
0 + 34247.32484r + r2 = 0 + 4096
34247.32484r + r2 = 0 + 4096

Combine like terms: 0 + 4096 = 4096
34247.32484r + r2 = 4096

The r term is 34247.32484r.  Take half its coefficient (17123.66242).
Square it (293219814.7) and add it to both sides.

Add '293219814.7' to each side of the equation.
34247.32484r + 293219814.7 + r2 = 4096 + 293219814.7

Reorder the terms:
293219814.7 + 34247.32484r + r2 = 4096 + 293219814.7

Combine like terms: 4096 + 293219814.7 = 293223910.7
293219814.7 + 34247.32484r + r2 = 293223910.7

Factor a perfect square on the left side:
(r + 17123.66242)(r + 17123.66242) = 293223910.7

Calculate the square root of the right side: 17123.78202092

Break this problem into two subproblems by setting 
(r + 17123.66242) equal to 17123.78202092 and -17123.78202092.

Subproblem 1

r + 17123.66242 = 17123.78202092 Simplifying r + 17123.66242 = 17123.78202092 Reorder the terms: 17123.66242 + r = 17123.78202092 Solving 17123.66242 + r = 17123.78202092 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-17123.66242' to each side of the equation. 17123.66242 + -17123.66242 + r = 17123.78202092 + -17123.66242 Combine like terms: 17123.66242 + -17123.66242 = 0.00000 0.00000 + r = 17123.78202092 + -17123.66242 r = 17123.78202092 + -17123.66242 Combine like terms: 17123.78202092 + -17123.66242 = 0.11960092 r = 0.11960092 Simplifying r = 0.11960092

Subproblem 2

r + 17123.66242 = -17123.78202092 Simplifying r + 17123.66242 = -17123.78202092 Reorder the terms: 17123.66242 + r = -17123.78202092 Solving 17123.66242 + r = -17123.78202092 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-17123.66242' to each side of the equation. 17123.66242 + -17123.66242 + r = -17123.78202092 + -17123.66242 Combine like terms: 17123.66242 + -17123.66242 = 0.00000 0.00000 + r = -17123.78202092 + -17123.66242 r = -17123.78202092 + -17123.66242 Combine like terms: -17123.78202092 + -17123.66242 = -34247.44444092 r = -34247.44444092 Simplifying r = -34247.44444092

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.11960092, -34247.44444092}

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